Inertial Theorem: Overcoming the quantum adiabatic limit

We present a theorem describing stable solutions for a driven quantum system. The theorem, coined inertial theorem, is applicable for fast driving, provided the acceleration rate is small. The theorem states that in the inertial limit eigenoperators of the propagator remain invariant throughout the dynamics, accumulating dynamical and geometric phases. The proof of the theorem utilizes the structure of Liouville space and a closed Lie algebra of operators. We demonstrate applications of the theorem by studying three explicit solutions of a harmonic oscillator, two-level and three-level system models. These examples demonstrate that the inertial solution is superior to that obtained with the adiabatic approximation. Inertial protocols can be combined to generate a family of solutions. The inertial theorem is then employed to extend the validity of the Markovian master equation to strongly driven open quantum systems. In addition, we explore the consequence of geometric phases associated with the driving parameters.